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An efficient algorithm for large-scale extended trust-region subproblems with non-intersecting linear constraints

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Abstract

In this paper, we study the extended trust-region subproblem in which the trust-region intersects the ball with m linear inequality constraints (m-eTRS). We assume that the linear constraints do not intersect inside the ball. We show that the optimal solution of m-eTRS can be found by solving one TRS, computing the local non-global minimizer of TRS if it exists and solving at most two TRSs with an additional linear equality constraint (1-eqTRS). Both TRS and (1-eqTRS) are polynomially and efficiently solvable, thus the new algorithm significantly improves over the SOCP/SDP relaxation of Burer and Yang [Math Program 149(1-2):253–264, 2015]. on two classes of test problems, the efficiency of the proposed approach is compared with the SOCP/SDP relaxation and branch and bound algorithm of Beck and Pan [J Global Optim 69(2):309–342, 2017].

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Notes

  1. Second order cone program/ Semidefinite program.

References

  1. Adachi, S., Iwata, S., Nakatsukasa, Y., Takeda, A.: Solving the trust-region subproblem by a generalized eigenvalue problem. SIAM J. Optim. 27(1), 269–291 (2017)

    Article  MathSciNet  Google Scholar 

  2. Beck, A., Eldar, Y.C.: Strong duality in nonconvex quadratic optimization with two quadratic constraints. SIAM J. Optim. 17(3), 844–860 (2006)

    Article  MathSciNet  Google Scholar 

  3. Beck, A., Pan, D.: A branch and bound algorithm for nonconvex quadratic optimization with ball and linear constraints. J. Global Optim. 69(2), 309–342 (2017)

    Article  MathSciNet  Google Scholar 

  4. Beck, A., Vaisbourd, Y.: Globally solving the trust Region subproblem using simple first-order methods. SIAM J. Optim. 28(3), 1951–67 (2018)

    Article  MathSciNet  Google Scholar 

  5. Bienstock, D., Michalka, A.: Polynomial solvability of variants of the trust-region subproblem. In: Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 380–390 (2014)

  6. Burer, S., Anstreicher, K.M.: Second-order-cone constraints for extended trust-region subproblems. SIAM J. Optim. 23(1), 432–451 (2013)

    Article  MathSciNet  Google Scholar 

  7. Burer, S., Yang, B.: The trust region subproblem with non-intersecting linear constraints. Math. Program. 149(1–2), 253–264 (2015)

    Article  MathSciNet  Google Scholar 

  8. Conn, A.R., Gould, N.I., Toint, P.L.: Trust Region Methods. SIAM, Philadelphia (2000)

    Book  Google Scholar 

  9. Fortin, C., Wolkowicz, H.: The trust region subproblem and semidefinite programming. Optim. Methods Softw. 19(1), 41–67 (2004)

    Article  MathSciNet  Google Scholar 

  10. Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 2.0 beta. http://cvxr.com/cvx (2013)

  11. Hsia, Y., Sheu, R.L.: Trust region subproblem with a fixed number of additional linear inequality constraints has polynomial complexity (2013). arXiv:1312.1398

  12. Jeyakumar, V., Li, G.Y.: Trust-region problems with linear inequality constraints: exact SDP relaxation, global optimality and robust optimization. Math. Program. 147(1–2), 171–206 (2014)

    Article  MathSciNet  Google Scholar 

  13. Locatelli, M.: Exactness conditions for an SDP relaxation of the extended trust region problem. Optim. Lett. 10, 1141–1151 (2016)

    Article  MathSciNet  Google Scholar 

  14. Martínez, J.M.: Local minimizers of quadratic functions on Euclidean balls 336 and spheres. SIAM J. Optim. 4(1), 159–176 (1994)

    Article  MathSciNet  Google Scholar 

  15. Moré, J.J., Sorensen, D.C.: Computing a trust region step. SIAM J. Sci. Stat. Comput. 4, 553–572 (1983)

    Article  MathSciNet  Google Scholar 

  16. Rendl, F., Wolkowicz, H.: A semidefinite framework for trust region subproblems with applications to large scale minimization. Math. Program. 77(1), 273–299 (1997)

    Article  MathSciNet  Google Scholar 

  17. Salahi, M., Fallahi, S.: Trust region subproblem with an additional linear inequality constraint. Optim. Lett. 10(4), 821–832 (2016)

    Article  MathSciNet  Google Scholar 

  18. Salahi, M., Taati, A.: A fast eigenvalue approach for solving the trust region subproblem with an additional linear inequality. Comput. Appl. Math. 37(1), 329–347 (2018)

    Article  MathSciNet  Google Scholar 

  19. Salahi, M., Taati, A., Wolkowicz, H.: Local nonglobal minima for solving large scale extended trust region subproblems. Comput. Optim. Appl. 66(2), 223–244 (2017)

    Article  MathSciNet  Google Scholar 

  20. Sturm, J.F., Zhang, S.: On cones of nonnegative quadratic functions. Math. Oper. Res. 28(2), 246–267 (2003)

    Article  MathSciNet  Google Scholar 

  21. Wang, A.L., Kilinc-Karzan, F.: The generalized trust region subproblem: solution complexity and convex hull results. arXiv:1907.08843

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

    S. Ansary Karbasy, M. Salahi & A. Taati

  2. Center of Excellence for Mathematical Modeling, Optimization and Combinatorial Computing (MMOCC), University of Guilan, Rasht, Iran

    M. Salahi

  3. Department of Mathematics, Statistics and Physics, College of Arts and Sciences, Qatar University, Doha, Qatar

    A. Hamdi

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  1. S. Ansary Karbasy
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  2. A. Hamdi
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  3. M. Salahi
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  4. A. Taati
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Correspondence to A. Hamdi.

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S. Ansary Krabasy: PhD Student.

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Karbasy, S.A., Hamdi, A., Salahi, M. et al. An efficient algorithm for large-scale extended trust-region subproblems with non-intersecting linear constraints. Optim Lett 15, 1425–1446 (2021). https://doi.org/10.1007/s11590-020-01652-6

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  • DOI: https://doi.org/10.1007/s11590-020-01652-6

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